一维囚禁Riesz气体线性统计量的累积量与大偏差
Cumulants and large deviations for the linear statistics of the one-dimensional trapped Riesz gas
AI总结：
本文研究一维囚禁Riesz气体线性统计量的涨落，推导了长短程相互作用下的累积量解析公式，得到其大偏差形式与“蒸发转变”，并将短程结果推广到指示函数，揭示了边缘处的标度行为。
摘要：
我们研究经典囚禁Riesz气体，即一维空间中位置为$x_i$的$N$个粒子，具有形式为$\boldsymbol{\frac{1}{|x_i-x_j|^{k}}}$（$k>-2$）的排斥幂律相互作用势，且处于形式为$V(x) \boldsymbol{\frac{1}{|x|^n}}$的外部约束势中。我们聚焦于该气体的平衡吉布斯态，其密度具有有限支撑区间$[-\boldsymbol{\frac{\boldsymbol{\frac{1}{2}}}{2}},\boldsymbol{\frac{\boldsymbol{\frac{1}{2}}}{2}}]$。我们研究大$N$极限下，光滑函数$f(x)$对应的线性统计量$\boldsymbol{\frac{1}{N}} = \boldsymbol{\frac{1}{N}} f(x_i)$的涨落。我们得到了一般$k>-2$情形下$\boldsymbol{\frac{1}{N}}$的累积量解析公式。对于长程相互作用（即$k<1$，包括对数气体（$k \to 0$）和库仑气体（$k =-1$）），我们针对单项式$f(x)= |x|^m$推导了这些累积量。对于短程相互作用（即$k>1$，包括Calogero-Moser模型，即$k=2$），我们计算了一般$f(x)$对应的$\boldsymbol{\frac{1}{N}}$的三阶累积量，以及单项式$f(x)= |x|^m$对应的任意阶累积量。我们还得到了$\boldsymbol{\frac{1}{N}}$的概率分布的大偏差形式，该形式展现出一种“蒸发转变”，即$\boldsymbol{\frac{1}{N}}$的涨落由最大的$x_i$的涨落主导。此外，在短程情形下，我们将结果推广到（非光滑的）指示函数$f(x)$，从而得到了区间$[-L/2,L/2]$内粒子数的全计数统计的高阶累积量。我们特别证明，当$L/2$趋近于气体的边缘，即$L/\boldsymbol{\frac{1}{2}} \to 1$时，这些累积量呈现出一种有趣的标度形式，我们将其与实轴上互补区间的空概率的大偏差联系起来。
英文摘要：
We consider the classical trapped Riesz gas, i.e., $N$ particles at positions $x_i$ in one dimension with a repulsive power law interacting potential $\propto 1/|x_i-x_j|^{k}$, with $k>-2$, in an external confining potential of the form $V(x) \sim |x|^n$. We focus on the equilibrium Gibbs state of the gas, for which the density has a finite support $[-\ell_0/2,\ell_0/2]$. We study the fluctuations of the linear statistics ${\cal L}_N = \sum_{i=1}^N f(x_i)$ in the large $N$ limit for smooth functions $f(x)$. We obtain analytic formulae for the cumulants of ${\cal L}_N$ for general $k>-2$. For long range interactions, i.e. $k<1$, which include the log-gas ($k \to 0$) and the Coulomb gas ($k =-1$) these are obtained for monomials $f(x)= |x|^m$. For short range interactions, i.e. $k>1$, which include the Calogero-Moser model, i.e. $k=2$, we compute the third cumulant of ${\cal L}_N$ for general $f(x)$ and arbitrary cumulants for monomials $f(x)= |x|^m$. We also obtain the large deviation form of the probability distribution of ${\cal L}_N$, which exhibits an "evaporation transition" where the fluctuation of ${\cal L}_N$ is dominated by the one of the largest $x_i$. In addition, in the short range case, we extend our results to a (non-smooth) indicator function $f(x)$, obtaining thereby the higher order cumulants for the full counting statistics of the number of particles in an interval $[-L/2,L/2]$. We show in particular that they exhibit an interesting scaling form as $L/2$ approaches the edge of the gas $L/\ell_0 \to 1$, which we relate to the large deviations of the emptiness probability of the complementary interval on the real line.
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